Let be a nonsingular square matrix. Prove that if is a scalar multiple of an orthogonal matrix, then 2 Prove that if 2 then is a scalar multiple of an orthogonal matrix. You may assume the...

Let
be a nonsingular square matrix.

Prove that if

is a scalar multiple of an orthogonal matrix, then

₂

Prove that if

₂

then

is a scalar multiple of an orthogonal matrix. You may assume the singular value decomposition that says

where

and

are orthogonal and is a diagonal matrix of singular values.